PHY 4604 Fall 2010 – Homework 8
This assignment may be submitted for full credit until 5:00 p.m. on Monday,
December 13 (after which time no credit will be available). Please bring your
homework to NPB 2162, pushing it under the door if necessary.
Answer all six questions. Please write neatly and include your name on the front page of
your answers. You must also clearly identify all your collaborators on this assignment. To
gain maximum credit you should explain your reasoning and show all working.
As part of the Physics Department’s Academic Learning Compact, the final exam will include
two multiple-choice questions set by a professor other than the course instructor. These
questions will likely be simpler than typical PHY 4604 questions, but they may involve
numerical calculations. You will be allowed to use a calculator, and the exam sheet will
provide the values of any fundamental constants that you need. Questions 1–3 below are
examples of this type of multiple-choice problem. Both on this homework and on the exam,
partial credit will be available for incorrect answers supported by relevant reasoning.
1. How fast would an electron have to be moving for its de Broglie wavelength to be
250 nm? Assume that relativistic effects are negligible.
(a) 2.91 × 103 m/s
(b) 2.16 × 10−3 m/s
(c) 7.76 × 101 m/s
(d) 4.63 × 102 m/s
(e) 1.98 × 104 m/s
2. An electron is in a stationary state of an infinite spherical well described by the potential V (r) = 0 for r < a and V (r) = ∞ for r > a. Which one of the following formulae
could possibly describe the full dependence of the wave function on the azimuthal angle
φ?
(a) φ
(b) sin9 φ
(c) exp(−φ2 /2)
(d) 1/ cos2 φ
(e) exp(φ)
3. Which one of the following pairs consisting of an energy and a magnitude of orbital
angular momentum is possible for a hydrogen atom?
(a) -0.278 eV and 3.16 × 10−34 J·s.
(b) -0.439 eV and 2.58 × 10−34 J·s
(c) -0.850 eV and 4.71 × 10−34 J·s
(d) -0.544 eV and 3.65 × 10−34 J·s
(e) -0.378 eV and 1.82 × 10−34 J·s
4. A spinless particle has a wave function
ψ(r, θ, φ) ∝ r2 e−r/a
√
2 cos θ −
√
5 sin2 θ sin 2φ ,
where a is real and positive.
(a) What is the most probable value of the radial coordinate r?
(b) What is the expectation value of L2 , the magnitude squared of the orbital angular
momentum?
Hint: Figure out how to write the angular part of the wave function above, i.e.,
the term in the parentheses, as a linear combination of Ylm ’s. You should know
the value of L2 in each Ylm .
5. An electron of gyromagnetic ratio γ is placed in a uniform magnetic field of magnitude
B pointing along the positive z axis. There is no electric field, so the electron’s energy
is independent of spatial coordinate, i.e., the energy consists of a constant E0 associated
with the kinetic energy and the uniform potential V (r) plus a spin-dependent term.
At time t = 0, the electron’s spin is described by a spinor
1
i
,
χ(t = 0) = √
−2
5
1
0
where
and
are eigenvectors of the spin-z operator Ŝz with eigenvalues ~/2
0
1
and −~/2, respectively.
(a) Find the spinor χ(t) at a later time t > 0.
(b) Find the expectation value of Sx , Sy , and Sz at time t.
6. Griffiths Problem 4.55, parts (a)–(f) only. In part (e), give only the values of J 2 that
might be measured, and not the probability of each value.
1
0
The question refers to spinors χ+ =
and χ− =
, the same eigenstates of Ŝz
0
1
discussed in Question 5 above.